Frequency-domain stability language In loop-shaping and Bode analysis, the phase margin (PM) at the gain crossover frequency is equal to which expression?

Introduction / Context:Phase margin is a key robustness metric for single-loop feedback systems. It quantifies how far the loop phase is from the critical −180° at the frequency where the open-loop gain is unity (0 dB). Knowing the correct definition ensures consistent controller tuning and interpretation of Bode plots.

Given Data / Assumptions:

• Open-loop L(jω) = G(jω)H(jω) is stable and minimum-phase for standard definitions.
• Gain crossover frequency ω_gc is where |L(jω_gc)| = 1.
• Phase lag is measured as a positive number representing |∠L(jω)| below 0°.

Concept / Approach:By convention, PM = 180° + ∠L(jω_gc). If we express ∠L(jω_gc) as −(phase lag), then PM = 180° − (phase lag). For example, if the loop phase at crossover is −150°, the phase lag is 150°, and the phase margin is 30°. A positive PM indicates that additional phase lag (e.g., from delays) can be tolerated before instability occurs.

Step-by-Step Solution:

Verification / Alternative check:Nyquist interpretation gives the same result: PM is the angle by which the Nyquist curve at |L| = 1 falls short of the −1 point direction.

Why Other Options Are Wrong:

• Phase lag − 180° / +180° / +90°: Do not match the standard definition; they would yield incorrect margins.

Common Pitfalls:Confusing sign conventions or reading the phase at the wrong frequency (ensure it is at gain crossover, not phase crossover).

Final Answer:180° - phase lag